EC 8392 – DIGITAL ELECTRONICS UNIT – II - Rohini College ...

ROHINI COLLEGE OF ENGINEERING AND TECHNOLOGY

EC 8392 – DIGITAL ELECTRONICS


UNIT – II : COMBINATIONAL CIRCUIT DESIGN

MULTIPLEXER: (Data Selector)

A multiplexer or MUX, is a combinational circuit with more than one input line, one

output line and more than one selection line. A multiplexer selects binary information

present from one of many input lines, depending upon the logic status of the selection

inputs, and routes it to the output line. Normally, there are 2n input lines and n

selection lines whose bit combinations determine which input is selected. The

multiplexer is often labeled as MUX in block diagrams.

A multiplexer is also called a data selector, since it selects one of many inputs and

steers the binary information to the output line.

Fig : 2.30 - Block diagram of Multiplexer

2-to-1- line Multiplexer:

The circuit has two data input lines, one output line and one selection line, S.

When S= 0, the upper AND gate is enabled and I0 has a path to the output.

When S=1, the lower AND gate is enabled and I1 has a path to the output.



Fig : 2.31 - Logic diagram

The multiplexer acts like an electronic switch that selects one of the two sources.

Truth table:

S Y

0 I0

1 I1

4-to-1-line Multiplexer:

A 4-to-1-line multiplexer has four (2n) input lines, two (n) select lines and one output

line. It is the multiplexer consisting of four input channels and information of one of the

channels can be selected and transmitted to an output line according to the select

inputs combinations. Selection of one of the four input channel is possible by two

selection inputs.

Each of the four inputs I0 through I3, is applied to one input of AND gate. Selection lines

S1 and S0 are decoded to select a particular AND gate. The outputs of the AND gate are

applied to a single OR gate that provides the 1-line output.



Fig : 2.32 - 4-to-1-Line Multiplexer

Function table:

S1 S0 Y

0 0 I0

0 1 I1

1 0 I2

1 1 I3

To demonstrate the circuit operation, consider the case when S1S0= 10. The AND gate

associated with input I2 has two of its inputs equal to 1 and the third input connected to

I2. The other three AND gates have atleast one input equal to 0, which makes their

outputs equal to 0. The OR output is now equal to the value of I2, providing a path from

the selected input to the output.



The data output is equal to I0 only if S1= 0 and S0= 0; Y= I0S1‘S0‘. The

data output is equal to I1 only if S1= 0 and S0= 1; Y= I1S1‘S0.

The data output is equal to I2 only if S1= 1 and S0= 0; Y= I2S1S0‘.

The data output is equal to I3 only if S1= 1 and S0= 1; Y= I3S1S0. When these

terms are ORed, the total expression for the data output is,

Y= I0S1’S0’+ I1S1’S0 +I2S1S0’+ I3S1S0.

As in decoder, multiplexers may have an enable input to control the operation of the

unit. When the enable input is in the inactive state, the outputs are disabled, and when

it is in the active state, the circuit functions as a normal multiplexer.

Quadruple 2-to-1 Line Multiplexer:

This circuit has four multiplexers, each capable of selecting one of two input lines.

Output Y0 can be selected to come from either A0 or B0. Similarly, output Y1 may have

the value of A1 or B1, and so on. Input selection line, S selects one of the lines in each

of the four multiplexers. The enable input E must be active for normal operation.

Although the circuit contains four 2-to-1-Line multiplexers, it is viewed as a circuit that

selects one of two 4-bit sets of data lines. The unit is enabled when E= 0. Then if S= 0,

the four A inputs have a path to the four outputs. On the other hand, if

S=1, the four B inputs are applied to the outputs. The outputs have all 0‘s when E= 1,

regardless of the value of S.



Fig : 2.33 – Quadruple 2 to 1 MUX

Application:

The multiplexer is a very useful MSI function and has various ranges of applications in

data communication. Signal routing and data communication are the important

applications of a multiplexer. It is used for connecting two or more sources to guide to

a single destination among computer units and it is useful for constructing a common



bus system. One of the general properties of a multiplexer is that Boolean functions

can be implemented by this device.

Implementation of Boolean Function using MUX:

Any Boolean or logical expression can be easily implemented using a multiplexer. If a

Boolean expression has (n+1) variables, then ‗n‘ of these variables can be connected to

the select lines of the multiplexer. The remaining single variable along with constants 1

and 0 is used as the input of the multiplexer. For example, if C is the single variable,

then the inputs of the multiplexers are C, C‘, 1 and 0. By this method any logical

expression can be implemented.

In general, a Boolean expression of (n+1) variables can be implemented using a

multiplexer with 2n inputs.

1. Implement the following boolean function using 4: 1

multiplexer,

F (A, B, C) = ∑m (1, 3, 5, 6). Solution:

Variables, n= 3 (A, B, C) Select lines= n-

1 = 2 (S1, S0) 2n-1 to MUX i.e., 22 to 1 = 4

to 1 MUX

Input lines= 2n-1 = 22 = 4 (D0, D1, D2, D3)

Implementation table:

Apply variables A and B to the select lines. The procedures for implementing the

function are:

i. List the input of the multiplexer



ii. List under them all the minterms in two rows as shown below.

The first half of the minterms is associated with A‘ and the second half with A. The

given function is implemented by circling the minterms of the function and applying the

following rules to find the values for the inputs of the multiplexer.

1. If both the minterms in the column are not circled, apply 0 to the corresponding

input.

2. If both the minterms in the column are circled, apply 1 to the corresponding

input.

3. If the bottom minterm is circled and the top is not circled, apply C to the input.

4.If the top minterm is circled and the bottom is not circled, apply C‘ to the input.

Multiplexer Implementation:



2. F (x, y, z) = ∑m (1, 2, 6, 7) Solution:



3. F ( A, B, C) = ∑m (1, 2, 4, 5) Solution:

Variables, n= 3 (A, B, C) Select lines= n-

1 = 2 (S1, S0) 2n-1 to MUX i.e., 22 to 1 = 4

to 1 MUX

Input lines= 2n-1 = 22 = 4 (D0, D1, D2, D3) Implementation

table:




4. F( P, Q, R, S)= ∑m (0, 1, 3, 4, 8, 9, 15)

Solution:

Variables, n= 4 (P, Q, R, S) Select

lines= n-1 = 3 (S2, S1, S0)

2n-1 to MUX i.e., 23 to 1 = 8 to 1 MUX

Input lines= 2n-1 = 23 = 8 (D0, D1, D2, D3, D4, D5, D6, D7)





5. Implement the Boolean function using 8: 1 and also using 4:1

multiplexer F (A, B, C, D) = ∑m (0, 1, 2, 4, 6, 9, 12, 14)



Solution:

Variables, n= 4 (A, B, C, D) Select

lines= n-1 = 3 (S2, S1, S0)




Multiplexer Implementation (Using 8: 1 MUX):



Using 4: 1 MUX:

6. F (A, B, C, D) = ∑m (1, 3, 4, 11, 12, 13, 14, 15)

Solution:


lines= n-1 = 3 (S2, S1, S0)







7. Implement the Boolean function using 8: 1 multiplexer.

F (A, B, C, D) = A’BD’ + ACD + B’CD + A’C’D.

Solution:



Convert into standard SOP form,

= A‘BD‘ (C‘+C) + ACD (B‘+B) + B‘CD (A‘+A) + A‘C‘D (B‘+B)

= A‘BC‘D‘ + A‘BCD‘+ AB‘CD+ ABCD +A‘B‘CD + AB‘CD+A‘B‘C‘D+ A‘BC‘D

= A‘BC‘D‘ + A‘BCD‘+ AB‘CD + ABCD +A‘B‘CD +A‘B‘C‘D+ A‘BC‘D

= m4+ m6+ m11+ m15+ m3+ m1+ m5

= ∑m (1, 3, 4, 5, 6, 11, 15)





8. Implement the Boolean function using 8: 1 multiplexer.

F (A, B, C, D) = AB’D + A’C’D + B’CD’ + AC’D.

Solution:

Convert into standard SOP form,

= AB‘D (C‘+C) + A‘C‘D (B‘+B) + B‘CD‘ (A‘+A) + AC‘D (B‘+B)

= AB‘C‘D+ AB‘CD+ A‘B‘C‘D + A‘BC‘D +A‘B‘CD‘ + AB‘CD‘ +AB‘C‘D+ ABC‘D= AB‘C‘D

+ AB‘CD+ A‘B‘C‘D + A‘BC‘D +A‘B‘CD‘ + AB‘CD‘+ ABC‘D = m9+ m11+ m1+ m5+

m2+ m10+ m13

= ∑m (1, 2, 5, 9, 10, 11, 13).

Implementation Table:




9. Implement the Boolean function using 8: 1 and also using 4:1 multiplexer F (w, x, y,

z) = ∑m (1, 2, 3, 6, 7, 8, 11, 12, 14)

Solution:

Variables, n= 4 (w, x, y, z)

Select lines= n-1 = 3 (S2, S1, S0)






Multiplexer Implementation (Using 8:1 MUX):



(Using 4:1 MUX):

10. Implement the Boolean function using 8: 1 multiplexer

F (A, B, C, D) = ∏m (0, 3, 5, 8, 9, 10, 12, 14) Solution:


lines= n-1 = 3 (S2, S1, S0)









11. Implement the Boolean function using 8: 1 multiplexer

F (A, B, C, D) = ∑m (0, 2, 6, 10, 11, 12, 13) + d (3, 8, 14)

Solution:


lines= n-1 = 3 (S2, S1, S0)


Input lines= 2n-1 = 23 = 8 (D0, D1, D2, D3, D4, D5, D6, D7) Implementation

Table:




12. An 8×1 multiplexer has inputs A, B and C connected to the selection inputs S2, S1,

and S0 respectively. The data inputs I0 to I7 are as follows I1=I2=I7= 0; I3=I5= 1; I0=I4= D

and I6= D'.

Determine the Boolean function that the multiplexer implements.



F (A, B, C, D) = ∑m (3, 5, 6, 8, 11, 12, 13).



DEMULTIPLEXER:

Demultiplex means one into many. Demultiplexing is the process of taking information

from one input and transmitting the same over one of several outputs.

A demultiplexer is a combinational logic circuit that receives information on a single

input and transmits the same information over one of several (2n) output lines.

Fig : 2.34 - Block diagram of demultiplexer

The block diagram of a demultiplexer which is opposite to a multiplexer in its operation

is shown above. The circuit has one input signal, ‗n‘ select signals and 2n output signals.

The select inputs determine to which output the data input will be connected. As the

serial data is changed to parallel data, i.e., the input caused to appear on one of the n

output lines, the demultiplexer is also called a ―data distributer‖ or a ―serial-to-

parallel converter‖ .



1-to-4 Demultiplexer:

A 1-to-4 demultiplexer has a single input, Din, four outputs (Y0 to Y3) and two select

inputs (S1 and S0).

Fig : 2.35 - Logic Symbol

The input variable Din has a path to all four outputs, but the input information is

directed to only one of the output lines. The truth table of the 1-to-4 demultiplexer is

shown below.

Enable S1 S0 Din Y0 Y1 Y2 Y3

0 x x x 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 1 1 0 0 0

1 0 1 0 0 0 0 0

1 0 1 1 0 1 0 0

1 1 0 0 0 0 0 0

1 1 0 1 0 0 1 0



1 1 1 0 0 0 0 0

1 1 1 1 0 0 0 1

Truth table of 1-to-4 demultiplexer

From the truth table, it is clear that the data input, Din is connected to the output

Y0, when S1= 0 and S0= 0 and the data input is connected to output Y1 when S1= 0 and

S0= 1. Similarly, the data input is connected to output Y2 and Y3 when S1= 1 and S0= 0

and when S1= 1 and S0= 1, respectively. Also, from the truth table, the expression for

outputs can be written as follows,

Y0= S1’S0’Din

Y1= S1’S0Din

Y2= S1S0’Din

Y3= S1S0Din



Fig : 2.36 - Logic diagram of 1-to-4 demultiplexer

Now, using the above expressions, a 1-to-4 demultiplexer can be implemented using

four 3-input AND gates and two NOT gates. Here, the input data line Din, is connected

to all the AND gates. The two select lines S1, S0 enable only one gate at a time and the

data that appears on the input line passes through the selected gate to the associated

output line.

1-to-8 Demultiplexer:

A 1-to-8 demultiplexer has a single input, Din, eight outputs (Y0 to Y7) and three

select inputs (S2, S1 and S0). It distributes one input line to eight output lines based on

the select inputs. The truth table of 1-to-8 demultiplexer is shown below.



Din S2 S1 S0 Y7 Y6 Y5 Y4 Y3 Y2 Y1 Y0

0 x x x 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 1

1 0 0 1 0 0 0 0 0 0 1 0

1 0 1 0 0 0 0 0 0 1 0 0

1 0 1 1 0 0 0 0 1 0 0 0

1 1 0 0 0 0 0 1 0 0 0 0

1 1 0 1 0 0 1 0 0 0 0 0

1 1 1 0 0 1 0 0 0 0 0 0

1 1 1 1 1 0 0 0 0 0 0 0

Truth table of 1-to-8 demultiplexer

From the above truth table, it is clear that the data input is connected with one

of the eight outputs based on the select inputs. Now from this truth table, the

expression for eight outputs can be written as follows:

Y0= S2‘S1‘S0‘Din Y4= S2 S1‘S0‘Din

Y1= S2‘S1‘S0Din Y5= S2 S1‘S0Din

Y2= S2‘S1S0‘Din Y6= S2 S1S0‘Din Y3=

S2‘S1S0Din Y7= S2S1S0Din

Now using the above expressions, the logic diagram of a 1-to-8 demultiplexer can be

drawn as shown below. Here, the single data line, Din is connected to all the eight AND

gates, but only one of the eight AND gates will be enabled by the select input lines. For

example, if S2S1S0= 000, then only AND gate-0 will be enabled and thereby the data

input, Din will appear at Y0. Similarly, the different combinations of the select inputs, the

input Din will appear at the respective output.



Fig : 2.37 - Logic diagram of 1-to-8 demultiplexer

1. Design 1:8 demultiplexer using two 1:4 DEMUX.



2. Implement full subtractor using demultiplexer.

Inputs Outputs

A B Bin Difference(D) Borrow(Bout)

0 0 0 0 0

0 0 1 1 1

0 1 0 1 1

0 1 1 0 1

1 0 0 1 0

1 0 1 0 0

1 1 0 0 0

1 1 1 1 1



EC 8392 – DIGITAL ELECTRONICS UNIT – II - Rohini College ...

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